Differentiate. F(x)=(a x-b)/(c x-d) ; a, b, c, d constant. ...

Name: Problem 12, Chapter 3: Calculus: One Variable
Uploaded: 2021-07-22T00:52:40-08:00
Duration: 2 min 41 s
Channel: Kian Manafi
Description: Differentiate. F(x)=(a x-b)/(c x-d) ; a, b, c, d constant.

Key Concepts

Differentiation

Differentiation is the process of computing the derivative of a function, which represents the instantaneous rate of change of the function with respect to its variable. This concept is fundamental in calculus and is used to study rates, slopes, and the behavior of functions at various points.

Quotient Rule

The quotient rule is a technique used to differentiate functions expressed as the ratio of two differentiable functions. It states that if a function is given by f(x) = g(x)/h(x), then its derivative is f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2. This rule is particularly useful when dealing with rational functions where both the numerator and the denominator are differentiable.

Transcript

00:01 So we're going to be finding the derivative of ax minus b divided by cx minus d, and i just put all a, b, and c, and d in blue since they're constants.

00:09 And the way that we're going to find this derivative is using this power rule, or sorry, not the power rule, the quotient rule i have written up in the top right.

00:17 So we can say that this derivative is equal to the derivative of f of x.

00:22 So f of x is our numerator, so that's going to be the derivative of ax minus b, and then multiplied by g of x, where g of x is our d.

00:30 Denominator.

00:31 So that's cx minus d.

00:33 And then we have minus f of x, which is ax minus b, multiplied by the derivative of our denominator.

00:41 So the derivative of cx minus d.

00:44 And then we have divided by our denominator squared.

00:48 So cx minus d squared.

00:52 And so now we can see, really all we have to find are the derivatives of ax minus b and cx minus d.

00:59 So let's go ahead and do a x minus b first.

01:03 For this derivative we can cut it up into two different derivatives, the derivative of a x and then minus the derivative of b.

01:12 Since b is a constant, this second derivative is going to go to zero.